Exploring vortex structures in orbital-angular- momentum beams generated from planar geometric modes with a mode converter

It is theoretically demonstrated that the planar geometric mode with a π/2 mode converter, so called the circularly geometric mode, can be solved from the inhomogeneous Helmholtz equation by considering the pump distribution on the lasing mode. Theoretical analysis clearly reveal that the vortex structures of circularly geometric modes are determined by the minimum order of transverse lasing modes, the total number of transverse lasing modes and the degenerate condition in the cavity. Moreover, we experimentally manifest that the circularly geometric mode can be generated from the selective pumped solid-state laser with an external π/2 mode converter. To explore the vortex structures of the generated geometric modes, the interference patterns are performed by an experimental apparatus consisting of a Mach-Zehnder interferometer. 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Introduction
Over the past decades, the orbital angular momentum (OAM) or optical vortice has been exploited in a variety of applications, such as trapping [1,2] and rotating [3] of microscopic particles in hydrodynamics and biology, controlling the chirality of twisted metal nanostructures [4,5], quantum communication [6,7], and spiral interferometry [8]. Thus far the OAM can be generated from several special cases of laser modes with different techniques, such as generation of Laguerre-Gaussian (LG) modes [9], transformation from Hermite-Gaussian (HG) modes by lens coverters [10][11][12], and creation with Gaussian beams through spiral phase plates [13], synthesized holograms [14], spatial light modulation on liquid crystal cells [15] or dielectric wedges [16]. Nowadays, generation of light beams with huge OAM is an important and interesting task for potential applications including demonstration of optomechanical effects and trapping of cold atoms [17]. Therefore, the investigation of high-order laser modes is useful for developing the idea for generating the coherent structured light that can carry huge OAM.
Selective pumping is an interesting method to yield high-order transverse modes. Recently, the Herriott-type multipass beams, known as planar geometric modes, have been experimentally found under the selective pumping [18][19][20][21][22][23][24]. The lasing modes have a preference to be localized on the periodic ray trajectories when the cavity lengths are somewhat close to the degenerate cavities. More intriguingly, the planar geometric modes formed by the coherent superposition of HG eigenmodes could be converted into the corresponding LG modes by a π/2-cylindrical-lens mode converter for producing huge OAM. Although it has been verified that the planar geometric modes can be transformed into the circularly geometric modes by using a mode converter [25], there has been no study with regard to the vortex structures related to the circularly geometric modes thus far.
In this work we exploit the inhomogeneous Helmholtz equation to manifest a theoretical analysis of the circularly geometric modes performed by using the eigenmode expansion for considering the pump distribution on the lasing mode in Ref [26]. Through numerical calculations of the phase distribution, it can be found that the topological charge of the central singularity is determined by the minimum order of transverse lasing eigenmodes. Moreover, the number of singularities and the vortex distributions surrounding outside are decided by not only the minimum order of transverse lasing eigenmodes, but also the total number of transverse lasing eigenmodes and the degenerate condition in the cavity. In experiments, we use the selective pumped microchip laser with an external π/2-cylindrical-lens mode converter to generate the circularly geometric modes. To analyze vortex structures of generated geometric modes, the experimental equipment with a Mach-Zehnder interferometer is also employed to display the desired interference patterns. Note that the interference pattern is formed by the plane wave and the circularly geometric mode. It is found that the experimental interference pattern can be excellently reconstructed with the theoretical model. Furthermore, the experimental interference patterns formed by a tilted plane wave and a circularly geometric mode can agree very well with the numerical calculations. The good agreement confirms that an analysis of vortex structures for circularly geometric modes in the present theoretical model is trustworthy.

Theoretical analysis of vortex structures generated by a π/2 mode converter
Under the paraxial approximation, the eigenmodes for the laser cavity with a concave mirror at z = -L and a plane mirror at z = 0 can be expressed as the is the beam radius at the waist, , and λ is the emission wavelength. With the transformation of spatial morphologies, the corresponding , p LG  modes for the radial index p and the azimuthal index  can be expressed as: LG eigenmodes. With planar geometric modes expressed as the expansion of HG eigenmodes satisfying the inhomogeneous Helmholtz equation [26], it is intriguing to explore the spatial morphology of planar geometric modes by transforming each HG components into the corresponding LG modes. Under the selective pumping, the transformed modes, also called circularly geometric modes, can be expressed as: where o n is the central transverse order, o s is the central longitudinal order, z c is the location of the gain medium, γ represents the inverse quality factor, and η is a constant that includes the effective conversion efficiency and the overlap integral in the longitudinal direction. The eigenmode frequency of the concave-plano cavity can be given by , [ (   The promising manifestation not only accurately analyzes the vortex structures of circularly geometric modes but also provides a new attempt to employ the inhomogeneous Helmholtz equation to explore these structured beams.

Experimental results and discussions
To confirm the vortex structures of the circularly geometric modes in the experiment, we exploit the selective pumping to excite planar geometric modes in a concave-plano resonator as shown in Fig. 4. The laser medium is an a-cut 2.0-at.% Nd 3+ :YVO 4 crystal with a length of 2 mm. Both sides of the Nd:YVO 4 crystal are coated for antireflection at 1064 nm (reflection < 0.1%). The radius of curvature of the concave mirror is R = 20 mm and the reflectivity is 99.8% at 1064 nm. The output coupler is a flat mirror with a transmission of 2% at 1064 nm. The pump source is an 809 nm fiber-coupled laser diode with a core diameter of 100 μm, a numerical aperture of 0.16, and a maximum output power of 3 W. A focusing lens with 20 mm focal length and 90% coupling efficiency is used to reimage the pump beam into the laser crystal. The pump radius is estimated to be approximately 25 μm. For a 20-mm radius-ofcurvature concave mirror, it is found that the planar geometric mode could be realized when the optical cavity length L′ is designed to be 18.5 mm. At a pump power of 1.5 W, the emission power of the planar geometric mode with the off-axis displacement x Δ = 0.3 mm can be seen to be 20 mW. For generating the interference patterns between plane waves and circularly geometric modes, the output planar geometric modes can be split into two beams after passing through a lens with focal length 125 mm and the beam splitter (BS 1 ). One sub-beam passes through a pin-hole with the radius 50 μm and the mirror M 1 ; the other sub-beam passes through a π/2 mode converter consisting of two cylindrical lenses and the mirror M 2 . Note that the π/2cylindrical-lens mode converter can transform the planar geometric modes into the circularly geometric modes when the axes of the cylindrical lenses are parallel to one another but at 45 ±° to the principal axes of the planar geometric modes. The focal length of the cylindrical lenses is f = 25 mm; the distance is precisely adjusted to be 2 f for the operation of the π/2 converter. Interference patterns formed by recombining two sub-beams behind another beam splitter (BS 2 ) are imaged by a CCD camera. Because of the circularly geometric modes formed by the superposition of the azimuthalorder LG modes can be further expressed as: LG 0,2 LG 0,3 LG 0,4 LG 0,13 LG 0,9 LG 0,5 LG Fig , c z = 10 mm, z = 10 mm and λ = 1064 nm, the circularly geometric modes eigenmodes. Furthermore, we employed the selective pumped solid-state laser with an external mode converter to generate the circularly geometric modes, and further exploited the Mach-Zehnder interferometer to display the interference pattern between the plane wave and the circularly geometric mode to investigate its vortex structures. It was clearly found that the numerical calculations agree very well with the experimental observations. Finally, we verified experimental and numerical interference patterns for circularly geometric modes by using a tilted plane wave to manifest the reliability of theoretical exploration. In addition to laser resonators with the selective pumping, spatial light modulators (SLMs) are the universal, adjustable, and widely applicable schemes for generating structured beams by controlling the phase of light distribution within an optical system. The intimate analysis of phase distribution in circularly geometric modes are also predictably applied to flexible SLMs. It is believed that the present exploration of circularly geometric modes can be used to develop in numerous optical manipulation because of their particular vortex structures and huge OAM.

Funding
This work is supported by the Ministry of Science and Technology of Taiwan (Contract No. MOST 105-2628-M-009-004).